Multi-objective optimal designs Pareto optimal set

The objective function is defined as a weighted linear combination of the initial cost function and of the expected value of the LCC. Further, the response of the structural system is constrained in terms of the modes (i.e. most probable values) of the non-

stationary response IDR amplitude PDFs of every DOF of the hysteretic MDOF system.

The design variables are the dimensions of the square cross-section of the column elements. Columns’ cross-section dimensions for a given floor are assumed to be equal, and thus the vector of design variables has three components, one for every story. Next, assuming an initial design and boundary constraints

, where the optimization

problem takes the form

where the conflicting sub-objectives are normalized as

In this regard, takes the form

under the stochastic constraints

and

In Eq.(7.20) stands for the initial cost which is assumed to be directly proportional to the building structure weight; this includes the weight of the column elements plus the weight of the plates evaluated at the design variables vector ; is the expected value of the LCC, evaluated at the design variables vector . In Eq.(7.21)

is a vector of the modes (i.e. most probable values) of the non-stationary

response IDR amplitude PDFs of every DOF of the hysteretic MDOF system for the whole duration of the seismic excitation with intensity factor , evaluated at the design variables vector . The structure design service life is considered to be equal to fifty years while the discount ratio, , is taken to be equal to . Regarding the stochastic constraints of Eqs.(7.21) and (7.22) the critical excitation was selected to be the one with intensity factor yielding an earthquake input equal to ; see Fig(7.3). The rationale behind this choice lies in the fact that the above chosen value for

represents a relatively severe earthquake event which is characterized by a low

annual probability of occurrence according to the hazard curve depicted in Fig.(7.4); thus, highly appropriate for applying constraints considering safety issues (e.g., Porter, 2003; Fragiadakis et al. 2006). In this setting, the imposed stochastic constraint of Eq.(7.21) ensures that the vector of the modes of the non-stationary response IDR amplitude PDFs

of every DOF of the hysteretic MDOF system for the whole duration of the seismic excitation with intensity factor will not exceed a preselected limit which is taken equal to and corresponds to a specific damage state according to the defined IDR limits of Table 7.1.

Further, regarding the constraint of Eq.(7.22), it efficiently exploits one of the significant features of the approximate technique. Specifically, the technique not only provides with the system response amplitude PDF for each and every DOF, but also decouples the original -DOF system of Eq.(3.1) into SDOF LTV oscillators of the form given in Eq.(3.17) yielding time-varying effective stiffness and damping

constraint of Eq.(7.22) for avoiding “moving resonance” phenomena (e.g., Tubaldi and Kougioumtzoglou, 2014). In this regard, it facilitates the optimization process to avoid unnecessary optimal design searching in areas where surely optimal designs do not exist. Specifically, considering the quasi-stationary treatment of the LTV oscillator expressed by the following form

it can be reasonably argued that the maximum response variance of the original MDOF system occurs when the excitation EPS resonates with the LTV oscillator equivalent natural frequency . Thus, to avoid this resonance phenomenon, the constraint of Eq.(7.22) is formulated so that is kept outside a critical range in the frequency domain [ where the excitation EPS takes its largest values. In this regard, the expression

is adopted, where is a selected EPS value given as a percentage of the peak EPS value corresponding to the time instant where takes its peak value; see Figs.(7.3) and (7.12). In the herein considered application, was taken equal to

.

Note that the deterministic constraints of Eq.(7.23) ensure that the optimization procedure will provide applicable design solutions from a practical viewpoint. Further, the expected value of the total cost, the initial cost and the expected value of the LCC are related according to the following expression (e.g., Wen and Kang, 2001)

The Pareto front curves for both the expected value of the LCC and the expected value of the total cost with respect to the initial cost are presented in Fig.(7.13).

Figure 7.13. Pareto front curves for the expected values of LCC and total cost against the initial cost.

Next, to highlight the flexibility of the proposed methodology, the compromise design solution from the Pareto front curve exhibiting the lowest expected value of the total cost, as well as the ones corresponding to the two tails (see Fig.(7.13)) are presented in Table 7.2.

Designs x(m) Cin(x) Design A 1st 2nd 3rd 0.3892 0.3701 0.3294 Design B 1st 2nd 3rd 0.4750 0.4749 0.3981 Design C 1st 2nd 3rd 0.5492 0.5489 0.5471

Table 7.2. Synoptically presented results regarding three different design solution configurations from the Pareto front curves (Designs A, B and C).

Moreover it was deemed appropriate to present also the non-stationary response IDR amplitude PDFs determined by the analytical technique regarding the design variables vector that corresponds to the compromise solution named ''Design B''; see Figs.(7.14- 7.16). The presented results corresponds to the case where the imposed intensity factor is taken equal to .

Figure 7.14. Non-stationary response IDR amplitude PDF of the first DOF of the hysteretic MDOF system via the analytical approach (compromise solution-Design B).

Figure 7.15. Non-stationary response IDR amplitude PDF of the second DOF of the hysteretic MDOF system via the analytical approach (compromise solution-Design B).

Figure 7.16. Non-stationary response IDR amplitude PDF of the third DOF of the hysteretic MDOF system via the analytical approach (compromise solution-Design B).

In this setting, the designer/analyst possesses a considerable amount of information for every compromise solution configuration regarding the initial cost as well as the expected values of both the LCC and the total cost. This is of particular importance for an educated

decision-making analysis where the final optimal design will be the compromise solution that best balances the initial cost, the LCC cost, and the total cost according to the project stakeholders’ perspective.

Chapter 8

Concluding remarks

In this chapter, the main conclusions along with pertinent remarks associated with the analytical formulations and the numerical results considered in this thesis are presented and discussed. Also, potential directions for future research work are outlined.

In chapter 1 a conspectus of the objectives and tools of this thesis is provided as well as a brief review of methods for nonlinear stochastic dynamic analysis.

In chapter 2, various stochastic models for the representation of the seismic action are provided. These include phenomenological seismic stationary as well as non-stationary stochastic models of both the separable and non-separable form. Further, a seismological model (Boore, 2003) of the more sophisticated kind that is based on two basic parameters, namely the earthquake moment magnitude and the epicentral distance is also presented.

In chapter 3, a review of an alternative analytical/approximate method to the type of nonlinear stochastic dynamic analysis, recently proposed by Kougioumtzoglou and Spanos (2013) is given. The analytical approach based on the concepts of statistical linearization and of stochastic averaging has been developed for determining the evolutionary stochastic response of MDOF nonlinear systems.

In chapter 4 an approximate analytical technique for determining the time-varying survival probability and associated first-passage PDF of nonlinear/hysteretic MDOF structural systems subject to evolutionary stochastic excitation has been developed. Specifically, based on an efficient dimension reduction approach and relying on the concepts of stochastic averaging and statistical linearization, the original nonlinear n- degree-of-freedom system has been decoupled and cast into (n) effective SDOF LTV oscillators corresponding to each and every DOF. In this regard, time-varying effective

stiffness and damping elements corresponding to each and every DOF have been defined and computed, while the non-stationary marginal, transition and joint response amplitude PDFs have been efficiently determined in closed-form expressions. Finally, the MDOF system survival probability and first-passage PDF have been determined approximately in a computationally efficient manner. Overall, the developed technique exhibits enhanced versatility since it can handle readily a wide range of nonlinear behaviors as well as various stochastic excitations with arbitrary non-separable EPS forms that exhibit strong variability in both the intensity and the frequency content. A 3-DOF structural system exhibiting hysteresis following the Bouc-Wen model subject to evolutionary stochastic excitation of both separable and non-separble kind has been included in the numerical examples. Comparisons with pertinent Monte Carlo simulations have demonstrated the reliability of the technique. Future work may include adaptation of the proposed theoretical framework to count for reliability assessment of sensitive complex systems of engineering interest.

In chapter 5 a novel methodology for determining the seismic fragility of nonlinear

MDOF structural systems has been presented that can be potentially used in conjunction with a PBEE analysis framework. Specifically, fragility surfaces are determined for nonlinear/hysteretic MDOF structural systems subject to earthquake excitations compatible with a prescribed stochastic seismological model. Note that the employed vector-valued IM comprises two parameters, namely the earthquake moment magnitude (Mm) and the epicentral distance (r). The developed framework relies on an efficient

approximate dimension reduction/decoupling technique for determining the non- stationary system response amplitude PDFs based on the concepts of statistical linearization and of stochastic averaging; thus, computationally intensive Monte Carlo simulations are circumvented. Further, considering the inter-story drift ratio as the selected damage measure and appropriately defined damage states structural system related fragility surfaces are determined at a low computational cost as well.

This attribute renders the proposed methodology, hopefully, useful for efficient structural system fragility analysis and design applications, at least at a preliminary level.

A building structure comprising the versatile Bouc-Wen (hysteretic) model has served as a numerical example for demonstrating the reliability of the proposed fragility analysis methodology. Future work may stem from the combination of the chapters 4 and 5 by proposing an efficient fragility analysis framework regarding fragilities of the first- passage kind. The first-passage kind failure definition may, perhaps, be appropriate for the most severe damage state. The choice of this bound as a threshold for considering the first-passage problem is absolutely justified since the first violation of this barrier leads to a collapse. Considering hysteretic multi-story building structures and relying on the

proposed theoretical developments, fragility surfaces regarding first-passage kind fragilities could be obtained in a straightforward manner and at considerable low

computational cost.

In chapter 6 a framework for the efficient solution of structural robust optimization problems has been proposed which features controlling both inter-story drift and absolute floor acceleration non-stationary second order statistics. It can be viewed as a systematic and efficient methodology for providing optimal robust design solutions. Due to the joint consideration of displacement and acceleration constraints, the framework provides robust design solutions even in cases where potentially incurred damages are associated with non-structural components. An important feature of the proposed framework relates to the utilization of an efficient approximate frequency domain approach for determining the system response non-stationary second-order statistics; thus, circumventing computationally intensive MCS. The proposed stochastic structural design methodology can be used in a straightforward manner in structural design problems involving systems with a large number of DOFs and subject to stochastic earthquake excitation even of the non-separable kind.

Future work may include the extension of the herein stochastic design methodology to structural systems equipped with nonlinear energy dissipation devices. A potential direction for future work could include as well stochastic earthquake excitations of the non-separable kind which comprise some of the main characteristics of seismic shaking,

such as decreasing of the dominant frequency with respect to time (Liu, 1970; Spanos and Solomos, 1983).

In chapter 7 a performance-based multi-objective design optimization framework considering LCC has been developed for nonlinear/hysteretic MDOF structural systems subject to evolutionary stochastic excitations. Although the developments herein have been tailored specifically for earthquake engineering related applications, they can be readily modified to account for other hazard kinds as well. The developed framework relies on an efficient approximate dimension reduction technique for determining the non-stationary system response amplitude PDFs based on the concepts of statistical linearization and of stochastic averaging; thus, computationally intensive Monte Carlo simulations are circumvented. Note that the technique not only provides with the system response amplitude PDF for each and every DOF, but also decouples the original -DOF system into SDOF LTV oscillators yielding time-varying effective stiffness

and damping elements corresponding to each and every DOF. This important additional output has been exploited in the formulation of the optimization problem for avoiding “moving resonance” phenomena. Further, the framework can readily account for excitations with arbitrary non-separable EPS forms that exhibit strong variability in both the intensity and the frequency content.

In this regard, considering appropriately defined damage measures structural system related fragility curves for each story are determined at a low computational cost as well. Finally, the structural system design optimization problem is formulated as a multi- objective one to be solved by a Genetic Algorithm based approach; thus, various compromise solutions are obtained providing the designer with enhanced flexibility regarding decision-making analysis. A building structure comprising the versatile Bouc- Wen (hysteretic) model serves as a numerical example for demonstrating the efficiency of the proposed methodology. Future work may include the adaptation of the developed framework for the study of the advantageous contribution of passive vibration control devices such as tuned-mass-dampers, base isolators and viscous dampers on a realistic hysteretic multi-story building structure.

The proposed development contributes substantially to promoting well-established random vibration theory techniques in current problems related to the challenging area of nonlinear structural dynamics. Hopefully, such approaches will further contribute to familiarizing the structural engineering community with well-established and theoretically solid concepts from the random vibration field.

Appendix A

Spectral representation method for simulating time-histories as

samples of a stochastic process with a given power spectrum

Consider an one-dimensional, uni-variate, stationary, Gaussian stochastic process

with mean value equal to zero, autocorrelation function and two-sided power spectrum . The stochastic process can be simulated by the following series as where and with

In Eq.(A.4) represents an upper cut-off frequency beyond which the power spectrum may be reasonably assumed to be zero for either mathematical or physical reasons.

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